In common with many automatics, you cannot start the above engine unless either P or N is selected first, so there is a "switch" in the system. But what you have described sounds like the symptoms of a battery that does not have sufficient charge to turn the starter motor but just enough to operate the solenoid that engages the starter gear. Are the battery connecting terminals etc properly cleaned off and vaselined? It also could be that the starter solenoid/gear is sticking periodically and needs to be cleaned and lubricated as appropriate.

The starter solenoid is normally the smaller part at the side of the starter motor. You might check first of all that the solenoid trigger wire is intact and properly connected etc. I suspect you may need to have the starter motor removed to sort this out.